Question

The maximum and minimum magnitude of resultant of two vectors are 12 units and 6 units respectively. Then the magnitudes of two vectors will be

Answer 1

Answer:

Correct option is

D

13

Step 1: Equation for resultant of two vectors

Let θ be the angle between the vectors

A

and

B

The resultant of two vectors is given by

R=

A

2

+B

2

+2ABcosθ

For R

max

,θ=0

∘

⇒∣

R

max

∣=∣

A

∣+∣

B

∣

⇒17=∣

A

∣+∣

B

∣ ....(1)

For R

min

,θ=180

∘

⇒∣

R

min

∣=∣

A

∣−∣

B

∣

⇒7=∣

A

∣−∣

B

∣ ....(2)

Step 2: Calculation of Magnitude of

A

and

B

Adding equation (1) and (2) ⇒2∣

A

∣=24

⇒

A

=12

From equation (1)⇒ ∣

B

∣=5

Step 3: Calculation of R when vectors are perpendicular

∵

A

⊥

B

∣

R

∣=

A

2

+B

2

+2ABcos90

o

⇒∣

R

∣=

12

2

+5

2

=

169

units

⇒∣

R

∣=13 units

Hence the magnitude of resultant vector is 13 units. Option D is correct.

Answer 2

Answer:

The magnitude of vector A is equal to 12 units and the magnitude of vector B is equal to 3 units.

Explanation:

Given, the maximum magnitude of resultant vector, $R_{max}$ = 12 units

The minimum magnitude of resultant vector, $R_{min}$ = 6 units

Consider that θ is the angle between two vectors A and B:-

Then the resultant vector can be determined as-

$R = \sqrt{A^2+B^2+2ABcos\theta}$

When  θ = 0°, the maximum magnitude of resultant vector will be given by:-

$R_{max}=|\vec{A}|+|\vec{B}|$

$12=|\vec{A}|+|\vec{B}|$                                                          ...................(1)

When  θ = 180°, the minimum magnitude of resultant vector will be given by:-

$|R_{min}|=|\vec{A}|-|\vec{B}|$

$6=|\vec{A}|-|\vec{B}|$                                                           ...................(2)

Add the equation (1) and (2);

$2\;|\vec{A}|= 12+6$

$2\;|\vec{A}|= 18$

$|\vec{A}|= 9 \; units$

Substitute the value of magnitude of a in equation (1);

$12=9+|\vec{B}|$

$|\vec{B}|= 12-9$

$|\vec{B}| = 3\; units$

Therefore, the magnitude of vector A is equal to 12 and the magnitude of B is equal to 3.

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